Loop Permutations

Abstract

As explained in Section 1.4, an iteration-level loop transformation of the model loop nest L simply changes the given sequential (i.e., total) execution order of its iterations. Suppose the new execution order is also to be sequential. We can get a whole class of new sequential orders by permuting the loops. Originally, the points of the index space R of L were to be traced in the increasing (lexicographic) order of the index vector (I ₁, I ₂,..., I _m). Each permutation π of the set {1, 2,...,m} defines a new execution order of the iterations, where the index points are traced in the increasing (lexicographic) order of the vector (I _π(1), I _π(2),..., I _π(m)). The corresponding loop transformation is called a loop permutation. After a valid loop permutation, it may be possible to replace one or more do loops by their corresponding doall loops.